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Full text of "Treatise On Analysis Vol-Ii"

3    SPECTRUM OF A COMMUTATIVE BANACH ALGEBRA       323

Z is not empty: if Z were empty there would exist a finite open covering (Vj) of X,
and for each j a function xj e m such that */(/) = 0 for all t e Vj; now consider the
function ] Xj Xj , which belongs to m.)

(d) Deduce from (b) and (c) that the spectrum X(A) of the algebra of Section 15.1,
Problem 1 can be canonically identified with the interval [0, 1].

2. Show that the spectrum of the algebra A of Section 15.1, Problem 2 consists of a
single point, and that the unique maximal ideal of A is the radical of A (Section 15.2,
Problem 7).

3. In the algebra sf(X) of Section 15.1, Problem 3, show that \\xn\\ ^ ||*||B/(JI - 1)!, and
deduce that ^(X) is equal to its radical. Deduce that J/(X) has no characters (use
Problem 5 of Section 15.1).

4.    Let B be the Banach subalgebra of ^c^P) (notation of (15.3.8)) generated by

and the function |/0| : h-> ||. Show that X(B) is homeomorphic to the set of points
(*i, x2 , *3) e R3 such that x\ -f xl ^ x and 0 ^ *3 ^ 1. (Remark that B contains
all functions of the form i 0(1|), where g is a continuous function on [0, 1], and
that B also contains the subalgebra A0 of .^(D) introduced in (15.3.8). To show that
lx(/o)l ^ X(l/ol) for all characters x of B consider the function ( - (|| + e))"1,
which belongs to B for all e > 0.

5.    The space / (6.5) becomes a Banach algebra without unit element under the multipli-
cation (^nX7?*)  (nr)n)> Let A be the Banach algebra obtained by adjoining a unit
element to / (Section 15.1, Problem 5). Show that X(A) may be identified with the
compact subset of R consisting of +00 and the integers ^1 (use the fact that /
can be canonically identified with its dual). The Gelfand transformation ^A then
becomes the identity mapping, and ^A(A) is a nonclosed dense subalgebra of

(a) Show that the dual of the underlying Banach space of the Beurling algebra A
(Section 15.1, Problem 4) can be identified with the space of classes of A-measurable
complex-valued functions g such that

\\9\\2 = sup

where the supremum is taken over the set of functions w e Ci such that V(to) > 0.
Then \\g\\ is the norm on the dual A' of A. Show that

n   Zn -T- l
The canonical bilinear form </, #> (where /e A and g e A') is then equal to

(b)   Deduce from (a) that the characters of the Banach algebra A obtained by ad-
joining a unit element to A (Section 15.1, Problem 5) have as their restrictions to A) ^ 0 for all t e X, then the inverse x"1 of x in ^C(X) belongs to